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Four Intersections (Posted on 2024-09-03)

For which real numbers c is there a straight line that intersects the curve

y = x⁴ + 9x³ + cx² + 9x + 4

in four distinct points?

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

Comment 3 of 3 |

Let y = mx + b be the line

Then the x-coordinates of the four intersection points will be from the equation y = x^4 + 9x^3 + cx^2 + (9-m)x + (4-b)

This last quartic equation will necessarily have two relative minimums and a relative maximum, and two distinct inflection points.

This means that the conditions in Quartic Questions applies, specifically the second part of that problem.

Then 3b^2-8ac>0, and we know a=1 and b=9. Then 243/8>c is the answer.

Posted by Brian Smith on 2024-09-04 11:05:59

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